Linearization of Model Equations

In this chapter another usejul tool will be described for model analysis linearization of model equations. When combined with Laplace or z-transform, the process model can be decomposed into linear transfer functions and the dynamic response of the variable of interest can easily be obtained and compared with the solution from a simulation study. The advantage of working with linear transfer functions is that process variable interaction can be visualized and better understood than in the case of a simulation study. 

Many process models are non-linear. When the model consists of oidy one differential equation, the solution may not be too complicated However, when multiple differential equations are involved, it may be difficult to get an insight into the dynamic process behavior. Linearization of model equations can be a helpful tool for the following reasons  

Laplace transformation is another useful tool. When used in combination with linearization it can help us to write the model equations in transfer function format. Thus, to apply Laplace transform to non-Unear equations, they should first be linearized. 

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Kittrell et al. (1965a) also performed two types of estimation. First the data at each isotherm were used separately and subsequently all data were regressed simultaneously. The regression of the isothermal data was also done with linear least squares by linearizing the model equation. In Tables 16.7 and 16.8 the reported parameter estimates are given together with the reported standard error. Ayen and Peters (1962) have also reported values for the unknown parameters and they are given here in Table 16.9. 

The classification procedure developed by Madron is based on the conversion, into the canonical form, of the matrix associated with the linear or linearized plant model equations. First a composed matrix, involving unmeasured and measured variables and a vector of constants, is formed. Then a Gauss-Jordan elimination, used for pivoting the columns belonging to the unmeasured quantities, is accomplished. In the next phase, the procedure applies the elimination to a resulting submatrix which contains measured variables. By rearranging the rows and columns of the macro-matrix,... 

The multiple linear regression model is simply an extension of the linear regression model (Equation 12.7), and is given below ... 

As was noted in Section 2.1.1, the concentration oscillations observed in the Lotka-Volterra model based on kinetic equations (2.1.28), (2.1.29) (or (2.2.59), (2.2.60)) are formally undamped. Perturbation of the model parameters, in particular constant k, leads to transitions between different orbits. However, the stability of solutions requires special analysis. Assume that in a given model relation between averages and fluctuations is very simple, e.g., (5NASNB) = f((NA), (A b)), where / is an arbitrary function. Therefore k in (2.2.67) is also a function of the mean values NA(t) and NB(t). Models of this kind are well developed in population dynamics in biophysics [70], Since non-linearity of kinetic equations is no longer quadratic, limitations of the Hanusse theorem [23] are lifted. Depending on the actual expression for / both stable and unstable stationary points could be obtained. Unstable stationary points are associated with such solutions as the limiting cycle in particular, solutions which are interpreted in biophysics as catastrophes (population death). Unlike phenomenological models treated in biophysics [70], in the Lotka-Volterra stochastic model the relation between fluctuations and mean values could be indeed calculated rather than postulated. 

Principal component regression is typically used for linear regression models (Equation 6.7 or Equation 6.10), where the number of independent variables p is very large or where the regressors are highly correlated (this is known as multicollinearity). 

The theoretical and numerical basis of computational flow modeling (CFM) is described in detail in Part II. The three major tasks involved in CFD, namely, mathematical modeling of fluid flows, numerical solution of model equations and computer implementation of numerical techniques are discussed. The discussion on mathematical modeling of fluid flows has been divided into four chapters (2 to 5). Basic governing equations (of mass, momentum and energy), ways of analysis and possible simplifications of these equations are discussed in Chapter 2. Formulation of different boundary conditions (inlet, outlet, walls, periodic/cyclic and so on) is also discussed. Most of the discussion is restricted to the modeling of Newtonian fluids (fluids exhibiting the linear dependence between strain rate and stress). In most cases, industrial... 

Mathematical models of flow processes are non-linear, coupled partial differential equations. Analytical solutions are possible only for some simple cases. For most flow processes which are of interest to a reactor engineer, the governing equations need to be solved numerically. A brief overview of basic steps involved in the numerical solution of model equations is given in Section 1.2. In this chapter, details of the numerical solution of model equations are discussed. 

The simultaneous solution uses the Newton-Raphson method, which is based on linearizing the model equations. Two characteristics are inherent in this method. Since the equations are highly nonlinear, the success of linearization usually requires good starting values. On the other hand, as the solution is approached, the linearized equations become progressively more accurate and convergence is accelerated. 

Linear calibration model Equation for the instrumental response which is directly proportional to the concentration (of the form y = a + bx). (Section 5.3)... 

Even if isothermal behavior is assumed, there are still serious restrictions on the range of applicability of the method. Considering an instantaneous opening of the valve and linearizing the model equations it is possible to derive a simple analytic solution for the dimensionless pressure in the dosing... 

This so-called isotonic convection approximation results in a system of model equations, 94- 96, that are all linear and ean be solved analytically. With the same boundary conditions as Diamond and Bossert, Segel obtained for the reabsorbate tonicity... 

Very often in the literature short time solutions are needed to investigate the behaviour of adsorption during the initial stage of adsorption. For linear problems, this can be achieved analytically by taking Laplace transform of model equations and then considering the behaviour of the solution when the Laplace variable s approaches infmity. Applying this to the case of linear isotherm (eq. 9.2-3), we obtain the following solution for the fractional uptake at short times ... 

Taking fully into account these jumps in material characteristics in a multizone model in the time domain is a pain again the Laplace transform comes to the rescue and delivers computational elegance. In order to clarify this, we briefly return to our solution in the Laplace domain for a single zone, and express the concentration and flux values to the left of this zone in terms of the values to its right. The linearity of the equations guarantees that one can write... 

The Newton-Gauss method consists of linearizing the model equation using a Taylor series expansion around a set of initial parameter values bo, also called preliminary estimates, whereby only the first-order partial derivatives are consider... 

Based on the above assumptions, the model equations are shown in Table 4. The mass balance equations at the pellet and crystal level are based in the double linear driving model equations or bidisperse model[30]. The solution of the set of parabolic partial differential equations showed in Table 4 was performed using the method of lines. The spatial coordinate was discretized using the method of orthogonal collocation in finite elements. For each element 2 internal collocation points were used and the basis polynomial were calculated using the shifted Jacobi polynomials with weighting function W x) = (a = Q,p=G) hat has equidistant roots inside each element [31]. The set of discretized ordinary differential equations are then solved with DASPK solver [32] which is based on backward differentiation formulas. 

In Chapter 3, we covered the mathematical modeling of lumped systems as well as some preliminary examples of distributed systems using a systematic, generalized approach. The examples for distributed systems were preliminary and they were not sufficiently generalized. In this chapter, we introduce sufficient generalization for distributed systems and give more fundamentally and practically important examples, such as the axial dispersion model resulting in two-point boundary-value differential equations. These types of model equations are much more difficult to solve than models described by initial-value differential equations, specially for nonlinear cases, which are solved numerically and iteratively. Also, examples of diffusion (with and without chemical reaction) in porous structures of different shapes will be presented, explained, and solved for both linear and nonlinear cases in Chapter 6. 

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